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OP-9088OpenLogic and foundations

Erdős #598 · set theory / ramsey theory

Let mm be an infinite cardinal and κ\kappa be the successor cardinal of 2ℵ02^{\aleph_0}. Can one colour the countable subsets of mm using κ\kappa many colours so that every X⊆mX\subseteq m with ∣X∣=κ\lvert X\rvert=\kappa …

MSC 03 · Mathematical logic and foundations

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Problem statement

Let mm be an infinite cardinal and κ\kappa be the successor cardinal of 2ℵ02^{\aleph_0}. Can one colour the countable subsets of mm using κ\kappa many colours so that every X⊆mX\subseteq m with ∣X∣=κ\lvert X\rvert=\kappa contains subsets of all possible colours?

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The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #598 link
  2. [2]Upstream Lean 4 statement for Erdős #598 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 598 link

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